Iterative projection onto convex sets for quantitative susceptibility mapping

Purpose Quantitative susceptibility map (QSM) reconstruction is ill posed due to the zero values on the “magic angle cone” that make the maps prone to streaking artifacts. We propose projection onto convex sets (POCS) in the method of steepest descent (SD) for QSM reconstruction. Methods Two convex projections, an object‐support projection in the image domain and a projection in k‐space were used. QSM reconstruction using the proposed SD‐POCS method was compared with SD and POCS alone as well as with truncated k‐space division (TKD) for numerically simulated and 7 Tesla (T) human brain phase data. Results The QSM reconstruction error from noise‐free simulated phase data using SD‐POCS is at least two orders of magnitude lower than using SD, POCS, or TKD and has reduced streaking artifacts. Using the l1‐TV reconstructed susceptibility as a gold standard for 7T in vivo imaging, SD‐POCS showed better image quality comparing to SD, POCS, or TKD from visual inspection. Conclusion POCS is an alternative method for regularization that can be used in an iterative minimization method such as SD for QSM reconstruction. Magn Reson Med 73:697–703, 2015. © 2014 Wiley Periodicals, Inc.